Time Series Analysis, Forecasting and Control - Essay Example

Augmented Dickey-Fuller tests

The following table gives the results of the ADF test conducted in gretl;

The following hypothesis was to be tested using the ADF test;

• H0: δ = 0 (there is a unit root)
• HA: δ ≠ 0 (there is not a unit root)

The computed value of the ADF test is –1.782, its asymptotic p-value is greater than 5% (0.7137). Hence the null of nonstationarity cannot be rejected (there is a unit root).

Autocorrelation Function (ACF)

Autocorrelation function (ACF) is the cross-correlation of a signal with itself. It refers to the similarity between observations as a function of the time lag between them. ACF is a statistical tool for finding repeating patterns, for example, the presence of a periodic signal obscured by noise, or maybe identifying the missing fundamental frequency (Box and Jenkins, 1994) in a signal implied by its harmonic frequencies. 

Partial Autocorrelation Function (PACF)

Just like ACF, PACF plots (Box and Jenkins, 2008) are also commonly used tools for identifying the order of an autoregressive model. The partial autocorrelation of an AR (p) process is zero at lag p + 1 and greater. If the sample autocorrelation plot indicates that an AR model may be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. One looks for the point on the plot where the partial autocorrelations for all higher lags are essentially zero. Placing on the plot an indication of the sampling uncertainty of the sample PACF is helpful for this purpose: this is usually constructed on the basis that the true value of the PACF, at any given positive lag, is zero.

The figure below gives the ACF and PCF plots;

If the PACF displays a sharp cut-off while the ACF decays more slowly (i.e., has significant spikes at higher lags), we say that the stationaries series displays an "AR signature," meaning that the autocorrelation pattern can be explained more easily by adding AR terms rather than by adding MA terms. The plots clearly display a sharp cut-off for the PACF and a significantly reduced value of spikes for the ACF implying that we instead use an AR model.

Part 3

From the ACF and PACF plots above, it is clear that our best choice is an AR model and as such we present the ARMAX model. The model was estimated in Gretl using the Conditional Maximum Likelihood

Model 2: ARMAX, using observations 1981:10-2014:12 (T = 399)

Dependent variable: timeseriesdata

Coefficient

Std. Error

z

p-value

const

0.846273

0.322733

2.6222

0.00874

***

phi_1

0.992877

0.00598555

165.8792

<0.00001

***

theta_1

0.552444

0.0438309

12.6040

<0.00001

***

Timeseriesperiod

0.00434338

0.0035137

1.2361

0.21641

Mean dependent var

 162.7706

S.D. dependent var

 68.02335

Mean of innovations

−0.001507

S.D. of innovations

 1.036725

Log-likelihood

−580.5472

Akaike criterion

 1171.094

Schwarz criterion

 1191.039

Hannan-Quinn

 1178.994

Real

Imaginary

Modulus

Frequency

AR

Root 1

1.0072

0.0000

1.0072

0.0000

MA

Root 1

-1.8101

0.0000

1.8101

0.5000

LM test for autocorrelation up to order 12 -

 Null hypothesis: no autocorrelation

 Test statistic: Chi-square(10) = 83.8796

Estimation

In this section, we present the values of the forecasts for the last 10% data values;

269.93

276.93

283.96

291.03

270.62

277.63

284.67

291.74

271.32

278.33

285.37

292.44

272.02

279.03

286.08

293.15

272.72

279.74

286.78

293.86

273.42

280.44

287.49

294.57

274.12

281.14

288.20

295.28

274.82

281.85

288.90

295.99

275.52

282.55

289.61

296.70

276.22

283.26

290.32

297.41